On a Special Class of Fibrations and Kähler Rigidity

Paciﬁc Journal of Mathematics

ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY

NICKOLAS J.MICHELACAKIS

Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005

ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY

NICKOLAS J.MICHELACAKIS

Let ᏭᏮn be the class of torsion-free, discrete groups that contain a normal, at most n-step, nilpotent subgroup of finite index. We give sufficient conditions for the fundamental group of a fibration F → T → B, with base B an infra-nilmanifold, to belong to ᏭᏮn. Manifolds of this kind may, for example, appear as thin ends of nonpositively curved manifolds. We prove that if, in addition, we require that T be Kähler, then T possesses a flat Riemannian metric and the fundamental group π1(T) is necessarily a Bieberbach group. Further, we prove that a torsion-free, virtually polycyclic group that can be realised as the fundamental group of a compact, Kähler K(π, 1)-manifold is necessarily Bieberbach.

1. Introduction

Torsion-free, discrete, cocompact subgroups of the group of affine motions of ޒn were first studied by Bieberbach in 1912, and more recently by Charlap; they are called Bieberbach groups. They correspond precisely to the fundamental groups of compact manifolds endowed with a flat Riemannian metric [Charlap 1965], and such manifolds are finitely covered by flat tori [Bieberbach 1911]. L. Auslander [1960] and Lee and Raymond [1985] turned their attention to almost-Bieberbach groups, that is, torsion-free, discrete, cocompact subgroups of GoC, with C a maximal, compact subgroup of Aut G for G a simply connected, nilpotent Lie group. They succeeded in generalising much of Bieberbach’s work. Malcev’s equivalence [1949] shows that torsion-free, finitely generated, nilpotent groups correspond precisely to the fundamental groups of nilmanifolds, that is, compact manifolds of the form M = G/N, where G is a simply connected, nilpotent Lie group, and N a discrete subgroup. Theorem 3.2 shows that almost-Bieberbach groups correspond to infra-nilmanifolds, compact manifolds of the form G/ 0 with G as above and 0 a discrete subgroup of GoC, where MSC2000: primary 22E40; secondary 32Q15, 14R20. Keywords: affinely flat manifold, (almost)-crystallographic, (almost)-Bieberbach group, (almost)-torsion-free, (virtually) polycyclic group, nilpotent Lie group, discrete cocompact subgroups, lattice, Malcev completion, cohomology of groups, complex (Kähler) structure, group action, group representation, flat Riemannian manifold, (infra)-nilmanifold.

311 312 NICKOLASJ.MICHELACAKIS

C is a maximal compact subgroup of Aut G. We denote by ᏭᏮn the class of almost-Bieberbach groups whose maximal, normal, nilpotent subgroup is at most n-step nilpotent. We shall say that a group 0 admits an n-step almost-Bieberbach structure if and only if 0 ∈ ᏭᏮn and its maximal normal nilpotent subgroup is n-step nilpotent. (We know from [Gromov 1981] and [Wolf 1968] that, among finitely generated groups, virtually nilpotent groups are precisely those groups that have polynomial growth. For details and precise definitions, see those works or [Tits 1981].) We employ algebraic methods to study closed manifolds that fibre over infra- nilmanifolds. If F → T → B is such a fibration, where F, T and B are all acyclic, the long homotopy exact sequence reduces to a group extension of the form

1 −→ π1(F) −→ π1(T ) −→ π1(B) −→ 1.

Manifolds of this type appear as thin ends of geometrically finite hyperbolic manifolds, which are an interesting subclass of nonpositively curved manifolds. More specifically, Apanasov and Xie [1997] proved that if 0 ⊂ ᏴnoU(n−1) is a torsion- n−1 free discrete group acting on the Heisenberg group Ᏼn := ރ ×ޒ, the orbit space Ᏼn/ 0 is a Heisenberg manifold of zero Euler characteristic and a vector bundle over a compact manifold. Further, this compact manifold is finitely covered by a nilmanifold which is either a torus or a torus bundle over a torus. This gener- alises earlier results on almost flat manifolds concerning lattices in ᏴnoU(n − 1) [Gromov 1978; Buser and Karcher 1981]. As mentioned above, groups in ᏭᏮn correspond to infra-nilmanifolds. In Sec- tion 2 we study extensions of the form

1 −→ G −→ 0 −→ K −→ 1, with K ∈ ᏭᏮn, to provide sufficient conditions under which 0 belongs to ᏭᏮn. In particular, Proposition 2.2 guarantees the existence of an almost-Bieberbach structure on 0 provided G is a normal subgroup of 0 in a precise way. Proposition 2.4 does the same provided G lies in ᏭᏮn and the action of K on G respects some suitable minimal conditions. In Section 3 we use the Johnson–Rees characterisation of fundamental groups of flat, Kähler [Johnson and Rees 1991], and projective [Johnson 1990] manifolds, and apply the Benson–Gordon theorem [1988] for the existence of a Kähler structure on a compact nilmanifold to show, in Theorem 3.3, that the existence of a Kähler structure on a special fibration as above implies the existence of a flat Riemann metric on T . In particular, in ᏭᏮn, the classes of fundamental groups of Kähler and projective manifolds coincide, as shown in Corollary 3.4. Further, as a consequence of the Lefschetz hyperplane theorem and Bertini’s theorem, this ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 313 is a subclass of the class of fundamental groups of compact, closed, nonsingular projective surfaces. Finally, in Section 4, we use a structure theorem concerning virtually polycyclic groups, proved in [Dekimpe and Igodt 1994], together with the results in [Arapura and Nori 1999], to prove, in Theorem 4.2, that a torsion-free, virtually polycyclic group can be realised as the fundamental group of a K(π, 1)-compact, Kähler manifold if and only if it is Bieberbach of a special kind, namely, its operator homomorphism is essentially complex.

2. Group extensions

A group N is said to be nilpotent if its upper central series

1 = N0 C N1 = Z(N) C N2 C ··· , defined by Ni+1/Ni = Z(N/Ni ), is finite. If n is the smallest integer such that Nn = N, then N is said to be n-step nilpotent. We shall say that a finitely generated, torsion-free group 0 admits an (n-step) almost-Bieberbach group structure if it can be written as an extension of a finitely generated, (n-step) nilpotent group N by a finite group 8. Notice that, given such a torsion-free, finitely generated, nilpotent ∼ i group, its quotients Ni+1/Ni are of a special form, namely Ni+1/Ni = ޚ j . Lemma 2.1. Let 0 fit in an extension

p 0 −→ ޚm −→ 0 −→ G −→ 1, where the torsion-free group G has an n-step nilpotent, normal subgroup N of ﬁnite index and ޚm a trivial N-module. Then 0 ∈ ᏭᏮn+1.

Proof. Let G be deﬁned by the extension

1 −→ N −→ G −→ 8 −→ 1, with N n-step nilpotent, 8 ﬁnite and φ : 8 → Out N the operator homomorphism. Consider 0 := p−1(N). Then the extension

p (2–1) 0 −→ ޚm −→ 0 −→ N −→ 1 is central, which implies that 0 is at most (n+1)-step nilpotent. The proof is −1 ∼ m m ∼ completed by the observation that 0 = p (N)C0 and 0/0 = (0/ޚ )/(0/ޚ ) = ∼ m G/N = 8. Notice that 0 is torsion-free since so are ޚ and G. We now turn our attention to the ﬁbre of the ﬁbration F → T → B to prove the following: 314 NICKOLASJ.MICHELACAKIS

Proposition 2.2. Let 0 be a torsion-free extension

p (2–2) 1 −→ K −→ 0 −→ G −→ 1 of a finitely generated group K by a group G admitting an n-step nilpotent, almost- Bieberbach structure such that Im c is finite, where c : 0 → Aut K denotes the conjugation map. Then 0 ∈ ᏭᏮn+1. Proof. Since G admits an n-step almost-Bieberbach structure there is a short exact sequence 1 −→ N −→ G −→ 8 −→ 1 where N is n-step nilpotent, 8 is finite, and φ : 8 → Out N is the operator homomorphism. Let 0ˆ := p−1(N). Then 0ˆ fits in a short exact sequence

p 1 −→ K −→ 0ˆ −→ N −→ 1, where we denote by c¯ the restriction of the conjugation map c : 0 → Aut K to 0ˆ . Let 0 := Ker c¯, which is nonempty since 0 is inﬁnite. Then the extension

1 −→ 0 ∩ K −→ 0 −→ p(0) −→ 1 is central, with p(0) C N, and therefore itself nilpotent. This means that 0 ∩ K is a finitely generated, torsion-free, abelian group and 0 is at most (n+1)-step nilpotent. The proof is completed by observing that the normal subgroup 0 of 0ˆ ˆ has finite index in 0, since Im c¯ is finite. The group Aut K , for K a Bieberbach group, is not necessarily finite. For an example, see [Charlap 1986, p. 219]. It does, then, make sense to check what happens if the fibre admits an n-step almost-Bieberbach structure. But first: Proposition 2.3. Let 0 be a torsion-free extension

p 1 −→ K −→ 0 −→ ޚn −→ 0 of a Bieberbach group K by a free abelian group of rank n, such that ޚm ⊆ Z(0), where ޚm is the translation subgroup of K and Z(0) the center of 0. Then 0 ∈ ᏭᏮ2.

m m Proof. First observe that ޚ C 0, since ޚ ≤ Z(0). We therefore have a short exact sequence

p (2–3) 1 −→ K/ޚm −→ 0/ޚm −→ ޚn −→ 0 where K/ޚm is isomorphic to F, the ﬁnite holonomy group of K . We distinguish two cases: ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 315

n−1 n (i) Assume that (2–3) is a central extension. Then choose Q := (ޚ × kޚ) C ޚ of index k = |F|, with |F| the exponent of F. Let 00 := p−1(Q). Then 00 fits in a short exact sequence p 1 −→ F −→ 00 −→ Q −→ 0 0 m that splits as a direct product. By construction, 0 = (F × Q) C 0/ޚ is of finite index. Let q : 0 → 0/ޚm be the identification map. The free abelian group Q 0 m imbeds as a normal subgroup of 0 C 0/ޚ and so also, because Aut(F × Q) = Aut F×Aut Q, as a normal subgroup of 0/ޚm. Let 0˜ :=q−1(Q) and 0ˆ :=q−1(00); then Q =∼ 0/˜ ޚm and F × Q =∼ 0/ˆ ޚm. Since Q ≤ ޚn, it acts on ޚm in the same way as ޚn, namely trivially. So 0˜ is 2-step nilpotent normal in 0. One can further check that its index |0/0˜ | in 0 is finite, because |0/0˜ | = |0/0ˆ | · |0/ˆ 0˜ | = |ޚn/Q| · |F|. This completes the proof in this case.

(ii) Assume that the sequence (2–3) is not central, and let c : 0/ޚm → Aut F be the conjugation map. Since F is ﬁnite and 0/ޚm inﬁnite, the kernel of c is nontrivial. m Let 0 := Ker c C 0/ޚ , let F := F ∩ 0, and let Q := p(0). Then the extension 1 −→ F −→ 0 −→ Q −→ 0

n ∼ ρ with Q C ޚ (so that Q = ޚ for some ρ ≤ n) belongs to the previous case. The result now follows, since 0 has ﬁnite index in 0. Proposition 2.4. Let 0 be a torsion-free extension

p (2–4) 1 −→ K −→ 0 −→ G −→ 1, where K and G admit m-step and n-step almost-Bieberbach structures, respec- tively. If Z(L/Li ) ⊆ Z(0/Li ), where {Li }i is the upper central series of an m-step nilpotent, normal subgroup L of ﬁnite index in K , then 0 ∈ ᏭᏮn+m.

Proof. We ﬁrst check inductively that Li C 0. This is clear for i = 1. Assume it is true for i and let qi : L → L/Li be the identiﬁcation map, where

−1 Li+1 = q Z(L/Li ) , ∼ so that Li+1/Li = Z(L/Li ). Then Li+1/Li C Z(0/Li ) and Li+1 C 0. The rest of the proof also follows by induction, ﬁrst on m and then on n. The group G is of the form 1 −→ N −→ G −→ 8 −→ 1, where N is n-step nilpotent and 8 ﬁnite. By letting 0ˆ := p−1(N) we get a sequence

1 −→ K −→ 0ˆ −→ N −→ 1. 316 NICKOLASJ.MICHELACAKIS

The case m = n = 1 follows from Proposition 2.3. Assuming the theorem is true for some m and n = 1, we shall show it is true for m + 1 and n = 1. If K is of the form 1 −→ L −→ K −→ F −→ 1, where L is m-step nilpotent and F finite, consider L1 = Z(L). The conditions of ∼ ρ the theorem ensure that L1 = Z(L) = ޚ C 0 for some positive integer ρ. This gives a short exact sequence 1 −→ K/ޚρ −→ 0/ˆ ޚρ −→ ޚν −→ 0, ˆ ρ with ν > 0. Then {Li /L1}i is the upper central series of L/L1 and 0/ޚ admits an (n+1)-step almost-Bieberbach structure by the induction hypothesis. 0ˆ fits into a central short exact sequence 0 −→ ޚρ −→ 0ˆ −→ 0/ˆ ޚρ −→ 1. Lemma 2.1 now applies to prove that 0ˆ , and therefore 0, admit an (n+2)-step almost-Bieberbach structure. Now assume the theorem is true for all m and n up to a certain value. We complete the proof by showing it holds for all m and n + 1. If {Ni }i is the upper central series of some (n+1)-step nilpotent N, define −1 0 := p (Nn). Then 0 fits in

1 −→ K −→ 0 −→ Nn −→ 1 and admits an (n+m)-step almost-Bieberbach structure. Also there is a positive integer µ such that the sequence 1 −→ 0 −→ 0ˆ −→ 0/0ˆ =∼ ޚµ −→ 0 is exact. The induction argument on the ﬁbre implies that 0ˆ ∈ ᏭᏮn+m+1, and so n+m+1 0 ∈ ᏭᏮ too.

3. Almost-Bieberbach groups and Kähler structures

Let Aut G denote the group of automorphisms of a simply connected Lie group G. We shall be concerned with discrete subgroups 0 of Aut G that act properly discontinuously on G. A group 0 is said to be crystallographic if it is a cocompact, discrete subgroup of ޒnoO(n) ⊂ Aff(ޒn), where O(n) is the maximal compact subgroup of GL(n, ޒ) and Aff(ޒn) is the group of Euclidean motions of ޒn. It is a Bieberbach crystallographic group if it is torsion-free as well. Bieberbach groups are precisely the fundamental groups of compact, complete Riemannian manifolds that are flat (locally isometric to Euclidean space), as first proved in [Bieberbach 1911]. An alternative characterisation of flat Riemannian manifolds is that in such manifolds, ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 317 transition maps can be extended to elements of ޒnoO(n). Charlap [1965] classi- fied these manifolds, up to connection-preserving diffeomorphisms, by associating to a manifold M a short exact sequence

1 −→ 3 −→ G −→ 8 −→ 1 in which the holonomy group 8 of M is ﬁnite and 3 =∼ ޚn is the translation sub- ∼ n group of 0 = π1(M), a torsion-free, discrete, cocompact subgroup of ޒ oO(n) ⊂ Aff(ޒn). More generally, if G is a simply connected, nilpotent Lie group, we consider a maximal compact subgroup C ⊆ Aut G. A cocompact, discrete subgroup 0 of G o C is called an almost-crystallographic group, and if torsion-free it is called almost-Bieberbach. The quotient G/ 0 is called an infra-nilmanifold, and if 0 ⊆ G it is a nilmanifold. Most of Bieberbach’s work has been generalised to the nilpotent case in [Aus- lander 1960] and [Lee and Raymond 1985]:

Theorem 3.1 (Auslander). Let 0 ⊆ G o Aut G be an almost-crystallographic group, where G is a connected, simply connected, nilpotent Lie group. Then (0 ∩ G) C 0 is a cocompact lattice in G, and 0/(0 ∩ G) is ﬁnite. Parts of the statement of the following theorem can already be found in [Lee and Raymond 1985]. We simplify the proof. Theorem 3.2. 0 is almost-crystallographic if and only if it is of the form

1 −→ N −→ 0 −→ 8 −→ 1, with N ﬁnitely generated, torsion-free, maximal nilpotent, and 8 ﬁnite.

Proof. If 0 ⊆ Go Aut G is an almost-Bieberbach group, Theorem 3.1 says that N = 0∩G is a maximal nilpotent, normal subgroup of 0 of finite index, and finitely generated because it is a discrete subgroup of the nilpotent group G. To prove the converse, given an extension like the one in the statement of the theorem, with abstract kernel φ : 8 → Out N, consider the extension of the Malcev completion ᏺ of N, 1 −→ ᏺ −→ S(0) −→ 8 −→ 1, φ with abstract kernel ψ : 8 → Out N → Out ᏺ. The claim is that there is exactly → ˆ : → one extension of ᏺ by 8, namely ᏺ , ᏺoψˆ 8, where ψ 8 Aut ᏺ is a lifting morphism of ψ. Since Z(ᏺ) is a vector space and 8 is finite, H 3(8, Z(ᏺ)) vanishes and by [Mac Lane 1963, Theorem 8.7] the abstract kernel [8, ᏺ, ψ] has an extension. Furthermore, [Mac Lane 1963, Theorem 8.8] says that this extension is unique 318 NICKOLASJ.MICHELACAKIS because the set H 2(8, Z(ᏺ)) parametrizing all congruence classes of such extensions is null, for the same reason. So we know that there is precisely one extension ᏺ ,→ 0 →→ 8. If we can further show that ψ has a lifting morphism ψˆ : 8 → Aut ᏺ, ∼ then 0 = ᏺoψˆ 8. To this end, we apply induction on the nilpotency class of ᏺ. If ᏺ is 1-step nilpotent then ᏺ =∼ Z(ᏺ) and the result is obvious. If ᏺ is c-step nilpotent, consider the inverse image under the natural projection q : Aut ᏺ → Out ᏺ of the finite group ψ(8). This gives birth to a short exact sequence Inn ᏺ ,→ q−1(ψ(8)) →→ ψ(8) with Inn ᏺ =∼ ᏺ/Z(ᏺ) fulfilling the induction hypothesis. We can thus find a splitting morphism s : ψ(9) → q−1(ψ(8)) < Aut ᏺ. But now, s ◦ ψ is the lifting we were looking for, completing the proof. We thus have the commutative diagram

1 −→ N −→ 0 −→ 8 −→ 1     ıy y 1 −→ ᏺ −→ ᏺo8 −→ 8 −→ 1 The map , with  (n, g) = (ı(n), g), embeds 0 as a discrete, cocompact subgroup of the disconnected Lie group S(0), proving the theorem. Given a short exact sequence ޚ2n ,→ 0 →→ 8 with operator homomorphism φ : 8 → Aut ޚ2n, we say φ is essentially complex if there is a complex structure 2n 2n 2 for the 8-module ޚ ⊗ ޒ, that is, a map t ∈ Endޒ[8](ޚ ⊗ ޒ) such that t = −1. 2n 2n t In other words, φ : 8 → Aut ޚ is essentially complex if Im φ ⊆ GLރ (ޚ ⊗ޒ) , with 2n t 2n GLރ (ޚ ⊗ ޒ) := {m ∈ GLޒ(ޚ ⊗ ޒ) such that mt = tm}. Theorem 3.3. Let 0 be the torsion-free extension

1 −→ N −→ 0 −→ 8 −→ 1, where N is a torsion-free, finitely generated maximal nilpotent group and 8 is a finite group. Then there is a compact Kähler K(0, 1)-manifold M if and only if N =∼ ޚ2n and the operator homomorphism φ : 8 → Aut N is essentially complex. Proof. By Theorem 3.2 there is a connected, simply connected Lie group G such that 0 is a torsion-free, discrete, cocompact subgroup of Go Aut G. Since M is a K(0, 1)-manifold, its universal covering is homeomorphically equivalent to G and M =∼ G/ 0. The hypotheses on N say that G contains N ,→ 0 as a discrete cocompact subgroup. Then Mˆ =∼ G/N is a compact K(N, 1)-nilmanifold that covers M in a finite, unramified way. Because the Kähler condition is local, the fact that M admits a Kähler structure implies that Mˆ also admits a Kähler structure. The Benson–Gordon theorem says that this can happen only if N =∼ ޚ2n, forcing ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 319 the finite cover Mˆ of M to be holomorphically equivalent to the complex torus n 2n ރ /ޚ . The converse is settled by [Johnson and Rees 1991, Theorem 3.1].

Let Ꮾ᏷ be the class of groups that can be realised as fundamental groups of compact, Kähler manifolds whose underlying Riemannian structure is ﬂat, and Ꮾᏼ ⊆ Ꮾ᏷ the subclass consisting of groups that can be realised as fundamental groups of complex projective varieties. Let ᏭᏮ᏷ denote the class of groups that can be realised as fundamental groups of compact nilmanifolds; that is, compact manifolds of the form G/ 0, where G is a simply connected, nilpotent Lie group and 0 a discrete subgroup admitting a Kähler structure, and let ᏭᏮᏼ ⊆ ᏭᏮ᏷ be the subclass consisting of groups that can be realised as fundamental groups of complex projective nilvarieties.

Corollary 3.4. (1) ᏭᏮ᏷ ≡ Ꮾ᏷ ≡ Ꮾᏼ ≡ ᏭᏮᏼ.

(2) Every group in ᏭᏮ᏷ is the fundamental group of a smooth, compact, complex algebraic surface. Proof. (1) The ﬁrst equality follows directly from Theorem 3.3 and [Johnson and Rees 1991, Theorem 3.1]. The second is [Johnson 1990, Corollary 4.3], while the third stems from the ﬁrst two together with the inclusion ᏭᏮᏼ ⊆ ᏭᏮ᏷. (2) If M is a smooth projective manifold, then by Bertini’s theorem there is a smooth hyperplane section M(n−1). By the Lefschetz hyperplane theorem [Milnor 1963], πl (M, M(n−1))=0 for l

1 −→ π1(F) −→ π1(T ) −→ π1(B) −→ 1 of their respective fundamental groups satisﬁes the conditions of either Proposition 2.2 or Proposition 2.4, then T admits a ﬂat Riemannian metric.

4. Virtually polycyclic groups and Kähler rigidity

An affinely flat manifold is an n-manifold endowed with an atlas whose transition maps can be extended to elements of Aff(ޒn) = ޒnoGL(n, ޒ). A torsion-free group 0 is virtually polycyclic if it has a subgroup 00 of finite index which is polycyclic, that is, one that admits a finite composition series 00 ⊇ 01 ⊇ 02 ⊇ ∼ · · · ⊇ 0n = 1 such that 0i / 0i+1 = ޚ for all i. The number n is an invariant, called the rank of 0. Groups in ᏭᏮn are obviously virtually polycyclic. Auslander [1964] has conjectured that the fundamental group of a compact, complete, affinely 320 NICKOLASJ.MICHELACAKIS

flat manifold has to be virtually polycyclic. Milnor [1977] has shown that torsion- free, virtually polycyclic groups can be realised as fundamental groups of complete affinely flat manifolds. On the other hand, Johnson [1976] has proved that torsion- free, virtually polycyclic groups can be realised as fundamental groups of compact K(π, 1)-manifolds. However, contrary to the Bieberbach case, Benoist [1992] has given an example of a 10-step nilpotent group of rank 11, proving that it is not always possible to do both! If 0 is a virtually polycyclic group, the Fitting group of 0, denoted Fitt(0), is the unique maximal normal subgroup of 0. The closure Fitt(0) of the Fitting group of a group 0 is the maximal normal subgroup of 0 containing Fitt(0) as a normal subgroup of finite index. The basic property of Fitt(0) is that it leaves the quotient 0/Fitt(0) with no finite, normal subgroup in it — in other words, almost-torsion- free. In [Dekimpe and Igodt 1994] it is proved that if 0 is a finitely generated virtually nilpotent group then 0 is almost-torsion-free if and only if Fitt(0) is almost-crystallographic. If N is a torsion-free, finitely generated, c-step nilpotent group, then to any extension p N ,→ 0 →→ Q with abstract kernel ψ : Q → Out N we can inductively associate c morphisms ψi : Q → Aut(Ni+1/Ni ), where Ni+1/Ni = Z(N/Ni ). Now if q ∈ 0 is such Tc that p(q) has finite order in Q, and hq, Ni is nilpotent, then p(q) ∈ 1 Ker ψi . Tc Conversely, if q ∈ 0 is such that p(q) ∈ 1 Ker ψi , then hq, Ni is nilpotent in 0. We shall use the following lemma, which is half of [Dekimpe and Igodt 1994, Theorem 2.2]. For completeness, we write a proof here. Lemma 4.1. Let 0 be a virtually polycyclic group. If 0 is almost-torsion-free, Fitt(0) is torsion-free maximal nilpotent in 0. Proof. Since 0 is polycyclic-by-finite, Fitt(0) is finitely generated nilpotent. There- fore its torsion set is a finite characteristic subgroup of Fitt(0), and thus normal in 0, and hence trivial since 0 is almost torsion-free. So, 0 fits in an extension p (4–1) 1 −→ Fitt(0) −→ 0 −→ Q −→ 1 j with Fitt(0) torsion-free and Q abelian-by-finite, say A ,→ Q →→ F. Now let q ∈ 0 be such that N := hq, Fitt(0)i is nilpotent, and look at p(N). If p(N)∩A 6= {1} then −1 p (p(N) ∩ A) is normal in 0 since (p(N) ∩ A) C A is nilpotent as a subgroup of N. Thus, p−1(p(N) ∩ A) ⊆ Fitt(0) and p(N) ∩ A = {1}, a contradiction. We deduce that p(N) =∼ j(p(N)) ⊆ F, and hence that p(q) is of finite order in Tc Q. The discussion preceding the theorem shows that p(q) ∈ 1(ψi ) ∩ p(N), where ψi are the morphisms associated with (4–1), which is a finite group since ON A SPECIAL CLASS OF FIBRATIONS AND KÄHLER RIGIDITY 321

F is finite; therefore q ∈ Fitt(0). But since 0 is almost torsion-free, Fitt(0) is almost crystallographic and Fitt(Fitt(0)) = Fitt(0) is maximal nilpotent in Fitt(0), implying q ∈ Fitt(0), a contradiction. Theorem 4.2. Let 0 be a torsion-free, virtually polycyclic group. Then 0 can be realised as the fundamental group of a K(π, 1) compact, Kähler manifold if and only if 0 is Bieberbach with essentially complex operator homomorphism. Proof. The converse is the second half of Theorem 3.3. For the direct statement, observe that since 0 is torsion-free, it is almost-torsion-free. Thus, by Lemma 4.1, Fitt(0) is torsion-free maximal nilpotent in 0, and 0 fits in a short exact sequence of the form p 1 −→ Fitt(0) −→ 0 −→ Q −→ 1, where Q is abelian-by-finite. Since 0 is Kähler, by [Arapura and Nori 1999], there exists a nilpotent subgroup 1 ⊆ 0 of finite index. But 1 is necessarily contained in Fitt(0), so Q is finite, and Theorem 3.3 completes the proof with N = Fitt(0). Provided that the Auslander conjecture is true, Theorem 4.2 would immediately imply: Conjecture 4.3. If a Kähler manifold T is the total space of a fibration F → T → B where both the base B and the fibre F are infra-nilmanifolds, then T admits a Riemann flat structure.

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Received June 25 2003. Revised June 5 2004.

NICKOLAS J.MICHELACAKIS MATHEMATICS AND STATISTICS DEPARTMENT UNIVERSITYOF CYPRUS P.O. BOX 20537 NICOSIA 1678 CYPRUS [email protected]